Let C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a,b € R}. Define addition + and multiplication of such pairs by (u, v) + (x, y) = (u + x, v+y) and (u, v)⋅ (x, y) = (ux - vy, uy +vx) for all u, v, r, y € R. Together they form a triple < C, +, . >. (a) Show that multiplication is associative in . (b) Show that every element (a, b) E C has a negative, and every element (a, b) € C# has an inverse. (c) Prove or disprove: The system of real numbers R is isomorphic to the system Rx {0}, +, . >. Here, 0 ER is the zero of R. (d) True or false? Justify your answer: The triple < C, +, > must contain a subfield isomorphic to R.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a, b = R}. Define
addition + and multiplication of such pairs by
(u, v) + (x, y) = (u + x, v+y) and (u, v) • (x, y) = (ux — vy, uy + vx)
R. Together they form a triple < C, +, . >.
for all u, v, r, y
(a) Show that multiplication is associative in < C, +, . >.
(b) Show that every element (a, b) € C has a negative, and every element (a, b) € C# has an inverse.
(c) Prove or disprove: The system of real numbers R is isomorphic to the system < R× {0}, +, . >.
Here, 0 R is the zero of R.
(d) True or false? Justify your answer: The triple C, +, > must contain a subfield isomorphic
to R.
Transcribed Image Text:Let C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a, b = R}. Define addition + and multiplication of such pairs by (u, v) + (x, y) = (u + x, v+y) and (u, v) • (x, y) = (ux — vy, uy + vx) R. Together they form a triple < C, +, . >. for all u, v, r, y (a) Show that multiplication is associative in < C, +, . >. (b) Show that every element (a, b) € C has a negative, and every element (a, b) € C# has an inverse. (c) Prove or disprove: The system of real numbers R is isomorphic to the system < R× {0}, +, . >. Here, 0 R is the zero of R. (d) True or false? Justify your answer: The triple C, +, > must contain a subfield isomorphic to R.
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